Question

select the correct answer from each drop-down menu. consider this system of equations: -3x + 5y = 22 (equation 1) 20y - 11 = 12x (equation 2) this system of linear equations represents drop - down lines. the system of equation 1 and the equation 20y = 12x + 88 represents drop - down lines.

Explanation:

Step1: Analyze the first system

First, rewrite both equations in slope - intercept form ($y = mx + b$, where $m$ is the slope and $b$ is the y - intercept).

For equation 1: $- 3x+5y = 22$
Add $3x$ to both sides: $5y=3x + 22$
Divide by 5: $y=\frac{3}{5}x+\frac{22}{5}$

For equation 2: $20y-11 = 12x$
Add 11 to both sides: $20y=12x + 11$
Divide by 20: $y=\frac{12}{20}x+\frac{11}{20}=\frac{3}{5}x+\frac{11}{20}$

The slopes of the two lines are equal ($m=\frac{3}{5}$), but the y - intercepts ($\frac{22}{5}$ and $\frac{11}{20}$) are different. So, the first system represents parallel lines.

Step2: Analyze the second system

We have equation 1: $y=\frac{3}{5}x+\frac{22}{5}$ (from step 1)
And the new equation: $20y = 12x+88$
Divide by 20: $y=\frac{12}{20}x+\frac{88}{20}=\frac{3}{5}x+\frac{22}{5}$

The slope of the new line is $\frac{3}{5}$ (same as equation 1) and the y - intercept is also $\frac{22}{5}$ (same as equation 1). So, the second system represents coincident (same) lines.

Answer:

The first drop - down: parallel; The second drop - down: coincident (or same)